The method of indirect inference

To evaluate the models’ performance in fitting the real data this thesis uses the method o f indirect inference proposed in Minford, Theodoridis and Meenagh (2009)10.

The approach involves using an auxiliary model that is completely independent of the theoretical one to produce a description o f the data against which the performance of the theory is evaluated indirectly. Such a description can be summarised either by the estimated parameters o f the auxiliary model or by functions o f these; these are called the descriptors o f the data and are treated as the ‘reality’; the theoretical model being evaluated is then simulated to find its implied values for these.

Indirect inference has been widely used in the estimation o f structural models (e.g., Smith (1993), Gregory and Smith (1991, 1993), Gourieroux et al. (1993), Gourieroux and Monfort (1996) and Canova (2005)). Yet here a different use o f indirect inference is made as our aim is to evaluate models that are already calibrated. The common element is the use o f an auxiliary time series model. In estimation the parameters of the structural model are chosen such that when this model is simulated it generates estimates o f the auxiliary model similar to those obtained from the actual data. The optimal choices o f parameters for the structural model are those that minimise the distance between a given function o f the two sets o f estimated coefficients o f the auxiliary model. Common choices o f this function are the actual coefficients, the

10 See Meenagh, Minford and Wickens (2009) and Le, et al. (2009, 2011) for more applications of this approach. Le et al. (2011) deals with a wide range o f practical issues raised by this approach.

scores or the impulse response functions. In model evaluation the parameters of the structural model are taken as given. The aim is to compare the performance of the auxiliary model estimated on simulated data derived from the given estimates of a structural model—which is taken as a true model of the economy, the null hypothesis— with the performance of the auxiliary model when estimated from the actual data. If the structural model is correct then its predictions about the impulse responses, moments and time series properties o f the data should statistically match those based on the actual data. The comparison is based on the distributions of the two sets o f parameter estimates o f the auxiliary model, or o f functions of these estimates.

The testing procedure thus involves first constructing the errors implied by the previously estimated/calibrated structural model and the data. These are called the structural errors and are backed out directly from the equations and the data11. These errors are then bootstrapped and used to generate for each bootstrap new data based on the structural model. An auxiliary time series model is then fitted to each set of data and the sampling distribution o f the coefficients o f the auxiliary time series model is obtained from these estimates o f the auxiliary model. A Wald statistic is then computed to determine whether functions o f the parameters o f the time series model estimated on the actual data lie in some confidence interval implied by this sampling distribution.

Following Minford, Theodoridis and Meenagh (2009), this thesis takes a VAR(l) for the three macro variables (interest rates, output gap and inflation) as the appropriate auxiliary model and treats as the descriptors o f the data the nine VAR(l) coefficients and the three variances o f the involved variables. The Wald statistic is computed from these12. Thus effectively it is testing whether the observed dynamics and volatility o f the chosen variables are explained by the simulated joint distribution o f these at a given confidence level. The Wald statistic is given by:

11 Some equations may involve calculation of expectations. The method used here is the robust instrumental variables estimation suggested by McCallum (1976) and Wickens (1982): here the lagged endogenous data are set as instruments and the fitted values are calculated from a VAR(l)— this also being the auxiliary model chosen in what follows.

12 Note that the VAR impulse response functions, the co-variances, as well as the auto/cross correlations o f the left-hand-side variables will all be implicitly examined when the VAR coefficient matrix is considered, since the former are functions o f the latter.

(®-4>)’2 (M)_1(4>-®) [2.4]

where O is the vector of VAR estimates o f the chosen data descriptors just mentioned, with O and Z (00) representing, respectively, the mean and variance-covariance matrix o f these implied by bootstrap simulations13. This whole test procedure can be illustrated diagrammatically in Figure 2.1 as follows:

Figure 2.1: The Principle o f Testing using Indirect Inference

Panel A: coefficients (including two constants) in his vector of data descriptors. Here, since a VAR(1) is chosen to provide a parsimonious description o f the data and the models are tested against their capacity in fitting the data’s dynamic relations and size, the vector o f chosen data descriptors includes nine VAR(1) coefficients and the three data variances. No constant is included since demeaned de-trended data are used (See ‘data’ part in 2.2.2 below). Chapter 4 (4.3.1), for checking robustness, also extends the exercise to one in which differing orders o f VAR are tried. It turns out that the choice o f VAR order in the context here is really merely a matter of setting the test’s rejection power.

While the first panel in Figure 2.1 summarises the main steps o f the methodology just described, the ‘mountain-shaped’ diagram in panel B gives an example of how the

‘reality’ is compared to model predictions using the Wald test when two parameters of the auxiliary model are considered. Suppose the real-data estimates o f these are given at R and there are two models to be tested, each implies a joint distribution of these parameters shown by the ‘mountains’ ( a and/?). Since R lies outside the 95%

contour o f a , it would reject this model at 95% confidence level; the other model that generated /? would not be rejected, however, since R lies inside. In practice there are usually more than two parameters to be considered and henceforth the test is carried out by the Wald statistic o f [2.4].

The joint distribution mentioned above is a bootstrap distribution simulated from bootstrapping the innovations implied by the data and the theoretical model and it is therefore an estimate o f the small sample distribution14. Such a distribution is generally more accurate for small samples than the asymptotic distribution; it is also shown to be consistent by Le, et al. (2011) given that the Wald statistic is

‘asymptotically pivotal’; they also showed it had quite good accuracy in small sample Montecarlo experiments15.

2.2. 2 Data and calibration

Data

To test the Fed’s monetary policy in the Great Moderation this chapter employs the quarterly data published by the Federal Reserve Bank o f St. Louis from 1982 to 200716. Most discussions o f the Fed’s behaviour (especially those based on Taylor rules) are concerned with the periods that begin sometime in the 1980s but here 1982

14 The bootstraps in the tests here are all drawn as time vectors so that any contemporaneous correlation between the innovations will be preserved.

15 Specifically, they found that the bias due to bootstrapping was just over 2% at the 95% confidence level and 0.6% at the 99% level.

16 Data base o f Federal Bank o f St. Louis: http://research.stlouisfed.org/fred2/

is chosen as the starting point because many (including Bemanke and Mihov, 1998, and Clarida, Gali and Gertler, 2000) have argued that it was around then that the Fed switched from using non-borrowed reserves to setting the Funds rate as the instrument o f monetary policy. Taylor (1993) originally suggested a later starting point for his specification and plainly one could choose a variety o f different sample periods and test for that; a robustness check regarding this point is deferred to chapter 4.

The tests measure it as the deviation of current Fed rate from the steady-state value which is interpreted here as a linear trend (at a quarterly rate for compatibility with the quarterly inflation rate); output gap xt is approximated by the percentage deviation o f real GDP from its HP-trend value ; quarterly inflation nt is defined as17

the log difference between current CPI and the one captured in the last quarter. The data are also demeaned for simplicity. These are plotted in figure 2.2; the unit root test results are reported in table 2.2.

Figure 2.2: Demeaned-detrended Data o f Interest Rates, Output Gap and Inflation

.010. trend approximates the flexible-price output in line with the bulk o f other empirical work. To estimate the flexible-price output from the full DSGE model that underlies the three-equation representation here, one would need to specify that model in detail, estimate the structural shocks within it and fit the model to the unfiltered data, in order to estimate the output that would have resulted from these shocks under flexible prices. This is a substantial undertaking lying well beyond the scope o f this thesis, empirical representation of the output gap treats the output trend as a linear or HP trend instead o f the flexible-price output—this Taylor rule is used in the best-fitting ‘weighted’ models for both the full sample and the sample from 1984. Thus while in principle the output trend should be the flexprice output solution, it may be that in practice these models capture this rather badly so that it performs less well than the linear or HP trends.

Table 2.2: Unit Root Test for Stationarity

Time series 5% critical value ADF test statistics p-values*

X

^{-1.94 -2.8 0.0053}

x, -1.94 -2.95 0.0035

-1.94 -3.60 0.0004

Note: denotes the Mackinnon (1996) one-sided p-values; H0: the time series has a unit root.

Since the data are mean-deviations, a VAR(l) representation o f them contains no constant but only nine parameters in the autoregressive coefficient matrix. Also, the use o f such data requires dropping the constants in any equation o f the models as well.

This explains why the two transformed Taylor rules involved in model two and three have no constant at all.

Calibration

The values o f model parameters chosen are those commonly calibrated and accepted for the US economy in the literature. These are listed in table 2.3 as follows:

Table 2.3: Calibration of Model Parameters

P time discount factor 0.99

<J inverse of elasticity of intertemporal consumption 2

TJ inverse of elasticity of labour 3

CD Calvo contract price non-adjusting probability 0.53 G/Y steady-state government expenditure to output ratio 0.23

Y/C steady-state output to consumption ratio 1/0.77 (implied)

K ^ _ (1 - a>)(l - cofi) 0 42 (implied)

co

Y r = K(T1 + a L ) 2.36 (implied)

0 price elasticity of demand 6

a /y = Q 1 parameter driving the optimal timeless policy¹⁸ 1/6 (implied)

P degree of interest rates smoothness 0.76

r , interest rates response to inflation 1.44

r'x interest rates response to output gap 0.14

P v autoregressive coefficient of demand disturbance 0.91 (sample estimate)

p u» autoregressive coefficient of supply disturbance 0.82 (sample estimate) P i autoregressive coefficient of policy disturbance: model one 0.35 (sample estimate) P i autoregressive coefficient of policy disturbance: model two 0.37 (sample estimate) P i autoregressive coefficient of policy disturbance: model three 0.31 (sample estimate)

As table 2.3 shows, the quarterly time discount rate is calibrated as 0.99, implying an approximately 1% quarterly (or equivalently 4% annual) rate o f interest in steady state,

cr and t] are set to as high as 2 and 3 respectively as in Carlstrom and Fuerst (2008), who emphasized on the values’ consistency with the inelasticity o f both intertemporal consumption decision and labour supply shown by the US data. The Calvo price stickiness o f 0.53 and the price elasticity o f demand o f 6 are both taken from Kuester, Muller and Stolting (2009). These values imply a contract length o f more than three quarters19, while the constant mark-up o f price to nominal marginal cost is 1.2. The implied steady-state output-consumption ratio of 1/0.77 is calculated based on the steady-state ratio o f government expenditure over output o f 0.23 calibrated in Foley and Taylor (2004). The Taylor rule tested in model three follows the calibration in Carlstrom and Fuerst (2008), where the interest rates’ response to a unit change in inflation and output gap are 1.44 and 0.14 respectively, with the degree of

‘smoothness’ o f 0.76. The last five lines in table 2.3 also report the autoregressive coefficients o f the structural errors implied by the models, which are all sample estimates based on the real data20. Notice that both o f the demand and supply shocks

18 Nistico (2007) showed the relative weight a is equal to the ratio of the slope o f the Phillips curve to the price elasticity o f demand, namely, a = y / 6 .

19 To be accurate, 2(1 — (o)~x — 1 » 3 .26.

20 These estimates are all significant at 5% significance level.

are shown to be highly persistent, in contrast to the policy shocks reflected in all the three models.

2.2. 3 Evaluating the models’ performance—the test results

This section presents the test results for the three models considered; these are based on the VAR parameters and the data variances. Since there are three endogenous variables, the VAR(l) representation generates twelve components: the nine VAR coefficients and the three variances21. The tests calculate two kinds of Wald statistic (called ‘directed’ Wald statistics) according to the aspects o f the data the models are asked to fit: here the dynamics and the volatility o f the data. Another ‘full’ Wald statistic where the two properties are simultaneously considered is also calculated to measure the models’ overall performance. In both cases the Wald statistic is reported as a percentile, i.e. the percentage point where the data value comes in the bootstrap distribution. The test results in detail are as follows:

Model one (with optimal timeless policy)

Table 2.4 below summarises the test results regarding the dynamic properties of model one:

Table 2.4: Individual VAR Coefficients and the Directed Wald Statistic

VAR(l) 95% 95% Values estimated In/Out

Coefficients lower bound upper bound with real data

Pn

^{0.6454 0.9420 0.8017 In}

Pn

^{-0.0844 0.0439 0.0834 OuJ}

/?13 -0.1774 0.0991 0.0112 In

P2\

^{-0.2589 0.2578 -0.2711 Qui}

P

^{22 0.6685 0.9105 0.9009 In}

21 The VAR(l) is assumed as follows:

i,

^{" A i A l 2 a 3 "}

h

^{- 1}

x,

^{= A 2 2 / ^ 2 3}

_

^{0 3}

*

^{^ 3 3 .} _

A3

^{- 0.4037 0.1871 - 0.1090 In} corresponding 95% bounds implied by the theoretical model. Specifically, the response o f interest rates to the lagged output gap and the response o f output gap to the lagged interest rates, as well as the response o f inflation to the lagged output gap, are all shown to be more aggressive than what the theoretical model would predict. In particular, the interest rates’ response to the lagged output gap in reality is more than twice as great as what could be generated from model simulations. Overall, the directed Wald statistic is reported as 98.2; this indicates the model’s success in capturing the actual dynamics at the 99% confidence level, although it clearly fails at the more conventional 95% level. Clearly, all the DSGE models here have problems fitting the data closely; yet the main purpose here is to rank these and to see if one of these stands out as relatively acceptable.

Turning to the volatility o f the data, table 2.5 below shows the extent to which this is explained by the theoretical model:

Table 2.5: Volatility of the Endogenous Variables and the Directed Wald Statistic Volatility of the

Note: Values reported in table 2.5 are magnified by 1000 times as their original values.

As table 2.5 shows, not only are all three variances within their individual 95%

bounds but also the directed Wald percentile is 10.4. That is, at the confidence level of 95%, the observed volatility is not only individually, but also jointly explained by the theoretical model—with such a low Wald statistic, they are very close to the joint means o f the variances.

Note that by using the directed Wald the above have been examining the theoretical model’s partial capacities in explaining the data. To evaluate the model’s overall performance, however, the full Wald statistic needs to be calculated. This is reported in table 2.6 as 96.5; hence the null hypothesis that the theoretical model explains both the actual dynamics and volatilities is easily accepted at the 99% confidence level and only marginally rejected at 95%.

Table 2.6: The Full Wald Statistic

The concerned model properties Full Wald percentile

Dynamics + Volatility 96.5

To summarise, model one does not only provide a rough explanation for the actual dynamics, but also precisely captures the volatility shown by the real data; its overall fitness in explaining the data is fairly good as DSGE models go and we may consider it as a reasonable approximation to the real-world economy.

Model two (with the original Taylor rule)

Leaving the economic environment (i.e., the ‘IS’ and Phillips curves) unchanged, model two replaces the optimal timeless rule assumed in model one with the original Taylor rule, widely regarded as a good description o f the Fed’s monetary policy from the late 1980s until at least the early 1990s. The rule’s performance in mimicking the real dynamics for our sample chosen is reported as follows:

Table 2.7: Individual VAR Coefficients and the Directed Wald Statistic respectively. Overall, the directed Wald statistic is reported as 100, suggesting there is no hope at all for the theoretical model to generate a joint distribution o f the VAR coefficients that simultaneously explains the ones observed in reality. The theoretical model thus is totally rejected by the Wald test for the dynamics.

Yet the model can still explain the data volatility reasonably well, as shown in table 2.8. It generates slight excessive interest rates and inflation variances, but ideally matches series the variance o f the output gap. The directed Wald statistic for the variances is 91.5, comfortably accepted therefore at 95%.

Table 2.8: Volatility of the Endogenous Variables and the Directed Wald Statistic

Note: Values reported in table 2.8 are magnified by 1000 times as their original values.

Lastly, table 2.9 shows the full Wald statistic as 100. This is hardly surprising since it fails so badly to capture the dynamics of the data.

Table 2.9: The Full Wald Statistic

The concerned model properties Full Wald percentile

Dynamics + Volatility 100

Thus the results above suggest model two, where the original Taylor rule is set as the fundamental monetary policy, has only partially captured the characteristics shown by the data; unless the discussions are focused exclusively on the ‘size’ o f the economy’s fluctuations, such a model is not to be taken as a realistic description o f the prevailing economic reality.

Model three (with ‘interest-rates-smoothed’ Taylor rule [1.2])

In this last model a calibrated Taylor rule version whose specification reflects the feature of ‘interest rates smoothing’ is assumed to be the underlying policy reaction function. This rule suggests to set interest rates as a weighted average o f what was set in the last period and what would be required had the original Taylor rule been followed, with the weights being the degree o f ‘policy inertia’ and its complement, respectively. While ‘interest-rates-smoothed’ Taylor rules o f this sort are commonly claimed to be supported by empirical evidence as representing the Fed’s underlying

policy (e.g., Clarida, Gali and Gertler (1999, 2000), Rotemberg and Woodford (1997, 1998)), the test results o f this model version are revealed as follows:

Table 2.10: Individual VAR Coefficients and the Directed Wald Statistic VAR(l) model. Again, except for the output gap’s responses to the lagged interest rates and to its own lagged value, all dynamic relationships shown by the real data are individually captured by the simulated 95% bounds. Yet, the directed Wald statistic reported is as high as 99.9, indicating that the theoretical model can hardly be used for explaining the observed dynamics, as the set o f real-data-based estimates o f the VAR coefficients is not captured by the joint distribution o f these across model simulations, even at a 99% confidence level22.

Turning to the data volatility, table 2.11 shows the model has merely correctly mimicked the variance o f the output gap but has evoked too much for both the interest

22 These are rather similar to the results for model two.

rates and inflation; the directed Wald statistic is reported as 99.4, implying the model is not a proper explanation for the observed volatility, either.

Table 2.11: Volatility of the Endogenous Variables and the Directed Wald Statistic Volatility of the

Note: Values reported in table 2.11 are magnified by 1000 times as their original values.

Indeed, the poor explanatory power of model three is not only detected by the directed Wald statistics but also the full Wald statistic when its overall fit to data is evaluated:

Indeed, the poor explanatory power of model three is not only detected by the directed Wald statistics but also the full Wald statistic when its overall fit to data is evaluated:

In document US post-war monetary policy and the Great Moderation: was the Fed doing a good job? (Page 37-50)